There are two fuses in an electrical device. Let X denote the lifetime of the first fuse, and let Y denote the lifetime of the second fuse (both in years). Assume the joint probability density function of X and Y is

\(f(x, y)= \begin{cases}\frac{1}{6} e^{-x / 2-y / 3} & x>0 \text { and } y>0 \\ 0 & \text { otherwise }\end{cases}\)

a. Find \(P(X \leq 2 \text { and } Y \leq 3)\).

b. Find the probability that both fuses last at least 3 years.

c. Find the marginal probability density function of X.

d. Find the marginal probability density function of Y .

e. Are X and Y independent? Explain.

Equation Transcription:

Text Transcription:

f(x,y)={_0 otherwise^{1over 6}e^{-x/2-y/3} x>0 and y>0

P(X2 and Y3)

Solution:

Step 1 of 4 :

There are two fuses in an electrical device . Let X denote the lifetime of the first fuse, and let Y denote the lifetime of the second fuse (both in years). And the joint probability density function of X and Y is

We have to find

- P(X ≤ 2 and Y ≤ 3).
- The probability that both fuses last at least 3 years.

- The marginal probability density function of X.

- The marginal probability density function of Y.

- Check X and Y are independent.